44 ON THE EFFECT OF THE INTERNAL FRICTION OF FLUIDS If now we put X = XV"1, we shall have 0 and \r for the limits of X', and the second of these becomes infinite with r. Hence limit of F,(r) = (2r)-* e™ f \C" + D" log 2V) e-«*' X'^ (ZX' ...... (91). Jo r°° Now e"* af * dx = TT*, and if we differentiate both sides of the ' [ e-xaf-*dx = T(s) Jo with respect to s, and after differentiation put s = £, we get f e"*fl?-*logacte JO Putting x = mX; in these equations we get 'o equation r Jo f *e-«x' x'-* log V d\' = m-* {F (|) - TT*log m}, Jo where that value of m~* is to be taken which has its real part positive. Substituting in (91) we get limit of FJr) = (^-V emr { C" + (TT-* T'| - log ~") D"} . 3V ; \2<mrJ \ \ 2 6 2J j Comparing with (88) we get 30. We are now enabled to find the relation between C and jD arising from the condition that the motion of the fluid shall not become infinitely great at an infinite distance from the cylinder. The determination of the arbitrary constants A, B, C, D will present no further difficulty, We must have 5 = 0, since other- wise the velocity would be finite at an infinite distance, and then the two equations (83), combined with the relation above men- tioned, will serve to determine A, (7, D, The motion of the fluid will thus be completely determined, the functions F^(r)y F8(r) being given by (84) and (87). When the modulus of mr is large, the series in (87), though ultimately hypergeometrically conver-